Understanding Injectivity and Surjectivity of the Function f(x) x (x ≥ 0) and f(x) -x (x

In the realm of mathematical analysis, understanding the properties of a function, such as injectivity (one-to-one) and surjectivity (onto), is crucial for comprehending its behavior and applicability in various mathematical models. This article will delve into the function defined as follows:

Is the function f(x) x (x ≥ 0) and f(x) -x (x

1. Injectivity: One-to-One

A function is said to be injective if each output value corresponds to exactly one input value. In other words, if for all a and b in the domain, fa fb implies a b. We will analyze the given function to determine if it possesses this property.

Case 1: a, b ≥ 0

For a and b both non-negative, the function behaves as f(a) a2 and f(b) b2. If f(a) f(b), then:

[a^2 b^2]

This equation holds true if and only if a b, because both a and b are non-negative. Therefore, in this case, the function is injective.

Case 2: a, b

For a and b both negative, the function behaves as f(a) -a2 and f(b) -b2. If f(a) f(b), then:

[-a^2 -b^2]

This simplifies to:

[a^2 b^2]

Again, since both a and b are negative, a b. Hence, in this case, the function is also injective.

Case 3: a ≥ 0 and b

For a non-negative and b negative, f(a) a2 and f(b) -b2. Since a2 is non-negative and -b2 is non-positive, it is impossible for a2 to equal -b2. Therefore, no such a and b exist that would make f(a) f(b) in this case.

Since in all cases, fa fb implies a b, the function is injective.

2. Surjectivity: Onto

A function is surjective if for every y in the codomain, there exists an x in the domain such that f(x) y. We will now determine if the function f(x) x (x ≥ 0) and f(x) -x (x

The range of f(x) is all real numbers. For x ≥ 0, f(x) x2 covers all non-negative real numbers. For x 2 covers all non-positive real numbers. Together, these cover the entire set of real numbers, (mathbb{R}).

Given any real number y, we can always find a real number x such that:

[f(x) y]

Precisely, if y ≥ 0, we can take x y. If y 2 -y. Thus, the function is surjective.

Conclusion

The function f(x) x (x ≥ 0) and f(x) -x (x injective and surjective. This property is significant in many areas of mathematics and serves as a benchmark for understanding the behavior of functions.

By examining the function's explicit definition and its graph, it becomes clear that the function is both one-to-one and onto. Any mathematician or analyst familiar with the properties of functions would recognize the importance of these characteristics.

For further reading on related topics, consider exploring the more general principles of mathematical functions, injective and surjective functions, and their implications for function analysis.

Key Takeaways:

Injective (One-to-One): Each output value corresponds to exactly one input value. Surjective (Onto): For every y in the codomain, there exists an x in the domain such that f(x) y. The function f(x) x (x ≥ 0) and f(x) -x (x